Another objection to the VNM approach (and to expected utility approaches generally), the St. Petersburg paradox, draws on the possibility of infinite utilities. The St. Petersburg paradox is based around a game where a fair coin is tossed until it lands heads up. At this point, the agent receives a prize worth 2n utility, where n is equal to the number of times the coin was tossed during the game. The so-called paradox occurs because the expected utility of choosing to play this game is infinite and so, according to a standard expected utility approach, the agent should be willing to pay any finite amount to play the game. However, this seems unreasonable. Instead, it seems that the agent should only be willing to pay a relatively small amount to do so. As such, it seems that the expected utility approach gets something wrong.

Various responses have been suggested. Most obviously, we could avoid the paradox ... by limiting agents' utilities to finite values. However ... these moves seem ad hoc. It's unclear why we should set some limit to the amount of utility an agent can receive.

This is incorrect. The VNM theorem says that the utility function assigns a real number to each lottery. Infinity is not a real number, so the VNM system will never assign infinite utility to any lottery. Limiting agents' utilities to finite values is not an ad hoc patch; it is a necessary consequence of the VNM axioms. See also http://lesswrong.com/lw/gr6/vnm_agents_and_lotteries_involving_an_infinite/